Applied Element Method (AEM) in Dynamic and Seismic Analysis Earthquake Engineering and Engineering Seismology

Introduction Overview of the AEM Dynamic analysis with the AEM

Prof dr Stanko Brčić, Mr Mladen Ćosić

Faculty of Civil Engineering University of Belgrade

Borsko Jezero, May 19-22, 2014.

S.Brčić, M.Ćosić Applied Element Method Introduction Numerical methods - FEM/AEM Overview of the AEM Comparrison: FEM - AEM Dynamic analysis with the AEM Applicability of the AEM Contents

1 Introduction Numerical methods - FEM/AEM Comparrison: FEM - AEM Applicability of the AEM 2 Overview of the AEM Springs in the AEM Discretization in the AEM Large deformation static analysis with AEM 3 Dynamic analysis with the AEM Diﬀerential equations of motion Seismic analysis Progresive collapse analysis Extreme Loadingr for Structures

S.Brčić, M.Ćosić Applied Element Method Introduction Numerical methods - FEM/AEM Overview of the AEM Comparrison: FEM - AEM Dynamic analysis with the AEM Applicability of the AEM Applied Element Method

Numerical methods Applied Element Method (AEM) - relatively new numerical method (15-20 years) Two general approaches in numerical methods: - Continuum based numerical methods (FEM, BEM, . . . ) - Discrete formulations (Rigid Body and Spring Models, Discrete Element Method, EDEM, . . . ) AEM evolved from Extended DEM and RBSM as the new discrete numerical method

Numerical methods 1st comprehensive formulation: H.Tagel-Din: “A new eﬃcient method for nonlinear, large deformation and collapse analysis of structures”, PhD thesis, Civ.Eng.Dpt., Univ. of Tokyo, 1998 (mentor Prof. Kimiro Meguro) 1st use of the name AEM: Meguro K. Tagel-Din H.: “Applied Element Method for Structural Analysis: Theory and Applications for Linear Materials”, Struct.Eng./Earthquake Eng., JSCE Vol 17, No 1, pp 21-35, 2000

Numerical methods Finite Element Method (FEM) - still the most widely used numerical method FEM simulations in all engineering fields (solid, fluid, acoustic, thermal, electro-magnetic, chemical engineering, . . . ) FEM applied in non-engineering fields too (medicine,. . . ) FEM applied in various interdisciplinary areas (fluid-structure, soil-structure, etc.)

S.Brčić, M.Ćosić Applied Element Method Introduction Numerical methods - FEM/AEM Overview of the AEM Comparrison: FEM - AEM Dynamic analysis with the AEM Applicability of the AEM Contents

Connectivity between elements In the FEM elements are connected at nodes All deformation is happening inside the element itself In the AEM elements are connected through the whole surface using springs Each spring is actually 3 springs: 1 spring for normal and 2 springs for shear deformations In the AEM deformation is outside elements (in surface springs) Elements in AEM are rigid bodies

S.Brčić, M.Ćosić Applied Element Method Introduction Numerical methods - FEM/AEM Overview of the AEM Comparrison: FEM - AEM Dynamic analysis with the AEM Applicability of the AEM Applied Elements

Transition from large elements to small elements Due to various reasons (stress concentration,. . . ), areas of ﬁner mesh are necessary Coarse and ﬁne mesh in FEM: special transition elements In the AEM transition elements are not necessary Connection between elements in AEM is realized through the surface springs, not nodal contact

FEM AEM

Partial element connectivity

FEM - connectivity and continuity at nodes: C0, C1 In the FEM partial connectivity may be achieved only with additional elements and new dofs In the AEM partial connectivity is easily achieved with surface springs In the AEM discontinuity between elements may be easily achieved (zero spring stiﬀness)

Numerical methods Problems with FEM in ﬁelds with discontinuities in material Simulations of crack initiation, development, propagation Special techniques in FEM for crack analysis (smeared crack approach, discrete crack approach,. . . ) FEM simulation of conditions that deﬁne initiation of structural collapse - YES However, progressive structural collapse with FEM - NO

S.Brčić, M.Ćosić Applied Element Method Introduction Numerical methods - FEM/AEM Overview of the AEM Comparrison: FEM - AEM Dynamic analysis with the AEM Applicability of the AEM Contents

Numerical methods AEM is the only numerical method which can acurately analyze, track and vizualize structural behavior through all 3 stages of the loading: - Small displacements (linear elastic, non-linear) - Large displacements (geometric & material change, Element separation) - Collision and collapse (including progresive collapse) The only software related to AEM: Extreme Loadingr for Structures (ELS)

S.Brčić, M.Ćosić Applied Element Method Introduction Springs in the AEM Overview of the AEM Discretization in the AEM Dynamic analysis with the AEM Large deformation static analysis with AEM Applied Element Method

Overview of the AEM Discretization of a structure into nodes, surrounded by a small volume elements (rectangular in 2D, parallelopiped in 3D) Each node in AEM has 3 or 6 dofs (2D or 3D) Elements are connected by normal and shear springs Springs are responsible for internal deformations and stress ﬁeld in the structure Discrete elements around the nodal points move as the rigid bodies, but the element assembly is deformable Structural deformation is represented by the spring deformation

Three types of springs in the AEM There are 3 types of springs in the AEM: - Matrix (material) springs - Reinforcement springs - Contact springs Linear and/or non-linear (normal and shear) springs are representing the material behavior For the RC structures each reinforcement bar is represented by the corresponding normal and shear springs Contact springs are generated when elements collide with each other or with ground

Automatic element contact and collision in AEM Contact or collision in AEM is detected automatically Elements in AEM are able to contact and separate, re-contact again without user intervention There are 3 types of contacts in the AEM: - Corner-to-Face - Edge-to-Edge - Corner-to-Ground

Automatic element separation in the AEM When the average strain at the element face reaches the separation strain, all springs at that face are removed Separation strain is the average strain along the element face corresponding to material properties After removal of springs adjacent elements are no longer connected, until collision occurs If the collision occurs, elements collide as the rigid bodies

Discretization in the AEM The structure is virtually divided into small elements (rectangular, 2D, or cubic, 3D) Elements have one nodal point in their centers Springs on element faces are connection between elements Each reinforcement is presented by the corresponding spring (each spring is, actually, 2 or 3 springs: normal and shear)

Discretization in the AEM At the location of reinforcement two pairs of springs are used: for concrete and for reinforcement bar Reinforcement spring and concrete spring have the same strain Due to stress conditions, separation between elements occurs because of failure of a concrete-spring before the reinforcement-spring Therefore, relative displacement between reinforcement and concrete may be taken into account automatically

Spring stiffness in the AEM With reference to previous figure, spring stiffness are: - Normal spring E T d K = n a - Shear spring G T d K = s a E and G are the modulus of elasticity and the shear modulus (of the material)

Spring stiﬀness in the AEM T is the thickness of the element, a the distance between element centroids (nodal points), i.e. the length of the spring, while d is the distance between adjacent springs Therefore, T d is the aﬀected area of the spring, while a is the length of the spring For the reinforcement spring, T d is replaced by the area of the reinforcement bar T a → Aa

Stiﬀness matrix in the AEM Stiﬀness matrix components corresponding to each dof are determined by assuming unit displacement in considered dof and by determining the corresponding force in the centroid of each element

For two elements in 2D, and dofs u1, u2, . . . , u6, the stifness matrix is 6 × 6: K K K = 11 12 K21 K22

where submatrices K12 and K21 are symmetric: K12 = K21

Stiﬀness matrix in the AEM

Elements of the submatrix K11 are given by kij = kji, (i, j = 1, 2, 3) Diagonal elements

2 2 k11 = sin (Θ + α) Kn + cos (Θ + α) Ks 2 2 k22 = sin (Θ + α) Ks + cos (Θ + α) Kn 2 2 2 k33 = L2 cos α Kn + L sin α Ks

Stiﬀness matrix in the AEM Oﬀ-diagonal elements

k12 = −L sin(Θ + α) sin(Θ + α) Kn

+ L sin(Θ + α) cos(Θ + α) Ks

k13 = L cos(Θ + α) sin α Ks − L sin(Θ + α) cos α Kn

k23 = L cos(Θ + α) cos α Kn + L sin(Θ + α) sin α Ks

where L, Θ, α are given in the previous ﬁgure, while Kn and Ks are the normal and shear spring stiﬀness

Stiffness matrix in the AEM Presented stiffness coefficients are due to only one pair of springs (normal and shear) The stiffness coefficients depend on the spring stiffness and the spring location The global stiffness matrix is obtained by summing up the stiffness matrices of all pairs of springs around each element The order of the global stiffness matrix is 3N or 6N, where N is the number of elements (i.e. the nodal points)

Linear static analysis in the AEM The governing equation in the linear elastic static analysis is

K u = f

Matrix K is the global stifness matrix, u is the unknown displacement vector and f is the applied load vector Small displacement theory Obtained results (for suﬃcient number of elements and springs) are equal to the theoretical results

Number of elements and springs in the AEM Various parametric analyses in order to determine the number of elements and the number of springs over the face of each element For framed structures the optimum is - ≈ 10 elements along the cross-sectional height - ≈ 10 springs over each element face More elements and more springs give better accuracy, but also much greater CPU time

Large deformation static analysis in the AEM Large deformation static analysis: geometric and material non-linearity The governing equation for large deformation analysis:

K ∆u = ∆f + Rm + Rg

- K is the nonlinear stiﬀness matrix, ∆u incremental displacement vector, ∆f incremental load vector

- Rm is the residual force vector due to cracking or incompatibitity between spring strains and stresses

- Rg is the residual force vector due to geometric changes in the structure due to loading

S.Brčić, M.Ćosić Applied Element Method Diﬀerential equations of motion Introduction Seismic analysis Overview of the AEM Progresive collapse analysis Dynamic analysis with the AEM Extreme Loadingr for Structures Contents

Dynamic linear/nonlinear analysis in the AEM The governing equation for large deformation dynamic analysis:

M ∆u¨ + C ∆u˙ + K ∆u = ∆f(t) + Rm + Rg

- M are C are the mass and damping matrices and K is the nonlinear stiﬀness matrix - ∆u¨, ∆u˙ and ∆u are incremental acceleration, velocity and displacement vectors - ∆f(t) is the incremental load vector

Dynamic linear/nonlinear analysis in the AEM

- Rm is the residual force vector due to cracking or incompatibitity between spring strains and stresses

- Rg is the residual force vector due to geometric changes in the structure due to loading

Model on the shaking table and numerical model

S.Brčić, M.Ćosić Applied Element Method